What are reasons my UV map doesn't show up in 3D view?
This looks quite obvious to me: the normals are inverted, thus the faces get culled. In english, the textured view is optimised by showing only one side of the faces. This way the inside of the mesh doesn't get rendered.
For some reason you have the faces pointing the wrong way around (ie. Blender thinks that the inside of your mesh is the outside and vice versa.). To fix this, in edit mode select all vertices and go to the Shading / UVs tab (allready open in your screen capture) and under Normals hit recalculate. If this didn't work, you can try to manually invert them by hitting Flip Direction. Now the surface should work.
I hope this helps.